It means to die. When you do the undertaker 'lays you out' to prepare for the funeral. Basically they prepare you for the funeraland dress you in the shroud and then you lay on a slab before being placed in the coffin.

lay me down in the grave. It refers to what happens to most of us when we have died. Others don't just leave us where we died, they move us somewhere and lay us down for the rest of time. We are beyond doing anything for ourselves, after all.